Others have answered for the commercial and chartered sectors, which is increasingly being strangled due to regulation. This answer will focus on the private sector.
Where are these announced?
The short answer is they are not announced, at least publicly like on a web site. Any kind of advertising or public communication kicks them into the charter or commercial spheres. When you know that a private pilot is planning to fly, you can ask if he wants to take you along. It's that simple.
Meeting people and making personal contact replaces the internet. I think it's OK to exchange emails with somebody who already knows you and met you personally or through a club, but they are not going to warm up to somebody out of the blue. Also in terms of 'where are these announced', some discretion can be in order. If you go around identifying your contacts by name on your Facebook page, it may not be appreciated and you're likely to find your emails unanswered after that.
If you know something is on a pilot's bucket list, like flying a Nanchang CJ-6 (a vintage two seater on many dream bucket lists), you can keep your eyes open for that kind of opportunity. Whatever gets you talking to pilots and in the air is key towards empty leg travel.
Another example... In 2018, there will be a competition to fly around the world in Bristol Bulldogs. For people who live on the route and are aware that Bulldogs need to land for maintenance and fuel at the drop of a hat, there's an opportunity to offer some hospitality and a quick tour of your city.
A really great calling card is the ability to speak different languages and to let people know you're happy to translate. I have also done pet-sitting, baby-sitting, driving, tour guiding, and cabin clean up/dish-washing. Having multiple passports also gives you a slight edge because you don't face visa issues.
How can one travel this way?
Some important things to point out about this...
Timing. If you haven't done it before, it's not going to happen overnight unless you're very lucky. It takes a long time to get oriented and to establish yourself as an enthusiast.
Flexibility. You don't always go exactly when you want and where you want. Instead, you go where the pilot is going and that may leave you with additional transportation to reach your destination - or even stranded for a while. For example, a flight into Heathrow or Gatwick would be extraordinarily rare; the general case is they are headed for a private airport like Cotswold or even as far out as Humberside. There are about 20 - 25 airports in the greater London area and home counties, but not every type of plane is cleared for their type of strip(s). On the plus side and if you are younger, being 'stranded' at some random field gives the opportunity to hang around the terminal and meet pilots who might be flying to some place else (note: I never went 'total gypsy' myself, but have spoken with people who managed to hop respectable distances over the course of say, a year; others have found that the demands of being flexible became unpleasant as they got older).
Regulations. It's getting worse and worse. Airpooler is now moribund because of US regulations. BlackJet was supposed to have Uber's business model, but now appears to be offering very expensive pay-only arrangements to avoid being shut down. And the EU has jumped in with stricter interpretation of Regulation 965/2014, which is making some pilots reluctant to do anything at all unless they know you well.
How to learn more about it? In comments, you wrote you're looking to know more about it...
The best way to find out more is to join local aviation clubs and read aviation magazines oriented to private pilots. A simple Google search 'aviation clubs near London' returned two pages of results. I am in two for example.
Source: Club event at Rochester Airport, Kent, fair use
For aviation magazines, I can suggest "Pilot Getaways" (mostly US audience), "Pilot Magazine (UK version)" (UK general interest), "Plane and Pilot", (US, mixes private and business, more technical).
Some other helpful entry points...
You can attend events and festivals, like the annual one at Shobdon Field where anybody can go and enjoy the event while expressing enthusiasm for the sport.
You can be a volunteer in Project Orbis, or raise donations in a challenge event. The Corporate Angel Network also takes in volunteers for data entry (with no direct path to flying) but leading to a wealth of knowledge about flight schedules, pilots, fractional owners, private car operators, airports, and all the rest.
Check out newly converted or upgraded airports like Conington. Check out their engineering and repair facilities, what kinds of certifications they offer. Even Wifi and broadband. It's all useful knowledge.
Finally, having some kit, like a Delorme Communicator or a Zulu class headset can be a tremendous help. Certainly the pilot will have his own, but it helps to establish you as a serious enthusiast who uses his own stuff.